Task 1
getwd()
## [1] "/Users/damodarpai/Documents/Labratories/Lab 7/Lab 7"
Task 2
mychisim<-function(n1=10,sigma1=3,mean1=5,iter=1000,ymax=0.1){ # adjust ymax to make graph fit
y1=rnorm(n1*iter,mean=mean1,sd=sigma1)# generate iter samples of size n1
data1.mat=matrix(y1,nrow=n1,ncol=iter,byrow=TRUE) # Each column is a sample size n1
ssq1=apply(data1.mat,2,var) # ssq1 is s squared
w=(n1-1)*ssq1/sigma1^2 #chi-sq stat
hist(w,freq=FALSE, ylim=c(0,ymax), # Histogram with annotation
main=substitute(paste("Sample size = ",n[1]," = ",n1," statistic = ",chi^2)),
xlab=expression(paste(chi^2, "Statistic",sep=" ")), las=1)
lines(density(w),col="Blue",lwd=3) # add a density plot
curve(dchisq(x,n1-1),add=TRUE,col="Red",lty=2,lwd=3) # add a theoretical curve
title=expression(chi^2==frac((n[1]-1)*s^2,sigma^2)) #mathematical annotation -see ?plotmath
legend("topright",c("Simulated","Theoretical"),col=c("Blue","Red"),lwd=4,lty=1:2,bty="n",title=title) # Legend #
invisible(list(w=w,summary=summary(w),sd=sd(w),fun="Chi-sq")) # some output to use if needed
}
chisq1 = mychisim(n1 = 10, mean1= 10, sigma1 = 4, iter= 1000,ymax = 0.2)
chisq2 = mychisim(n1 = 20, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.08)
chisq3 = mychisim(n1 = 100, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.06)
chisq4 = mychisim(n1 = 200, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.04)
chisq = mychisim(n1 = 10, iter= 1500, mean1= 20, sigma1 = 10)
hist(chisq$w , col=rainbow(15), xlab = expression(paste(chi^2,"Value", sep=" "),las=1))
Task 3
myTsim = function(n1=10,sigma1=3,mean1=5,iter=1000,ymax=0.1){ # adjust ymax to make graph fit
y1=rnorm(n1*iter,mean=mean1,sd=sigma1)# generate iter samples of size n1
data1.mat=matrix(y1,nrow=n1,ncol=iter,byrow=TRUE) # Each column is a sample size n1
sd1=apply(data1.mat,2,sd) # sd
ybar=apply(data1.mat,2,mean) # mean
w=(ybar-mean1)/(sd1/sqrt(n1)) #T stat
hist(w,freq=FALSE, ylim=c(0,ymax), # Histogram with annotation
main=substitute(paste("Sample size = ",n[1]," = ",n1," statistic = ",T," iterations= ",iter)),
xlab=expression(paste(T, "Statistic",sep=" ")), las=1)
lines(density(w),col="Blue",lwd=3) # add a density plot
curve(dt(x,n1-1),add=TRUE,col="Red",lty=2,lwd=3) # add a theoretical curve
title=expression(T==frac((bar(y)-mu),s/sqrt(n1))) #mathematical annotation -see ?plotmath
legend("topright",c("Simulated","Theoretical"),col=c("Blue","Red"),lwd=4,lty=1:2,bty="n",title=title) # Legend #
invisible(list(w=w,summary=summary(w),sd=sd(w),fun="T")) # some output to use if needed
}
myT1 = myTsim(n1 = 10, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.5)
myT2 = myTsim(n1 = 20, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.5)
myT3 = myTsim(n1 = 100, iter= 1000, mean1= 10, sigma1 = 4,ymax = 0.5)
myT4 = myTsim(n1 = 200, mean1= 10, sigma1 = 4, iter= 1000, ymax = 0.5)
T = myTsim(n1 = 10, iter= 1500, mean1= 20, sigma1 = 10,ymax = 0.5)
hist(T$w , col=rainbow(15), xlab = expression(paste(chi^2,"Value", sep=" "),las=1))
Task 4
mychisim2<-function(n1=10,n2=14,sigma1=3,sigma2=3,mean1=5,mean2=10,iter=1000,ymax=0.07,...){ # adjust ymax to make graph fit
y1=rnorm(n1*iter,mean=mean1,sd=sigma1)# generate iter samples of size n1
y2=rnorm(n2*iter,mean=mean2,sd=sigma2)
data1.mat=matrix(y1,nrow=n1,ncol=iter,byrow=TRUE) # Each column is a sample size n1
data2.mat=matrix(y2,nrow=n2,ncol=iter,byrow=TRUE)
ssq1=apply(data1.mat,2,var) # ssq1 is s squared
ssq2=apply(data2.mat,2,var)
spsq=((n1-1)*ssq1 + (n2-1)*ssq2)/(n1+n2-2) # pooled s squared
w=(n1+n2-2)*spsq/(sigma1^2)#sigma1=sigma2, Chi square stat
hist(w,freq=FALSE, ylim=c(0,ymax), # Histogram with annotation
main=substitute(paste("Sample size = ",n[1]+n[2]," = ",n1+n2," statistic = ",chi^2)),
xlab=expression(paste(chi^2, "Statistic",sep=" ")), las=1)
lines(density(w),col="Blue",lwd=3) # add a density plot
curve(dchisq(x,n1+n2-2),add=TRUE,col="Red",lty=2,lwd=3) # add a theoretical curve
title=expression(chi^2==frac((n[1]+n[2]-2)*S[p]^2,sigma^2)) #mathematical annotation -see ?plotmath
legend("topright",c("Simulated","Theoretical"),col=c("Blue","Red"),lwd=4,lty=1:2,bty="n",title=title) # Legend #
invisible(list(w=w,summary=summary(w),sd=sd(w),fun="Chi-sq")) # some output to use if needed
}
mychi2var1 = mychisim2(n1 = 10,n2=10, iter= 1000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 0.5)
mychi2var2 = mychisim2(n1 = 20,n2=10, iter= 1000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 0.1)
mychi2var3 = mychisim2(n1 = 50,n2=50, iter= 1000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 0.1)
mychi2var4 = mychisim2(n1 = 80,n2=50, iter= 1000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 0.1)
mychi2 = mychisim2(iter=1000)
hist(mychi2$w,col=rainbow(15), xlab = expression(paste(chi^2,"Value", sep=" "),las = 1))
Task 5
myTsim2<-function(n1=10,n2=14,sigma1=3,sigma2=3,mean1=5,mean2=10,iter=1000,ymax=0.5,...){
y1=rnorm(n1*iter,mean=mean1,sd=sigma1)# generate iter samples of size n1
y2=rnorm(n2*iter,mean=mean2,sd=sigma2)
data1.mat=matrix(y1,nrow=n1,ncol=iter,byrow=TRUE) # Each column is a sample size n1
data2.mat=matrix(y2,nrow=n2,ncol=iter,byrow=TRUE)
ssq1=apply(data1.mat,2,var) # ssq1 is s squared
ybar1= apply(data1.mat,2,mean)
ssq2=apply(data2.mat,2,var)
ybar2=apply(data2.mat,2,mean)
spsq=((n1-1)*ssq1 + (n2-1)*ssq2)/(n1+n2-2) # pooled s squared
w=((ybar1-ybar2)-(mean1-mean2))/sqrt(spsq*(1/n1+1/n2))#sigma1=sigma2, Chi square stat
hist(w,freq=FALSE, ylim=c(0,ymax), # Histogram with annotation
main=substitute(paste("Sample size = ",n[1]+n[2]," = ",n1+n2," statistic = ",T)),
xlab=paste(" T Statistic",sep=""), las=1)
lines(density(w),col="Blue",lwd=3) # add a density plot
curve(dt(x,n1+n2-2),add=TRUE,col="Red",lty=2,lwd=3) # add a theoretical curve
title=expression(T==frac((bar(Y)[1]-bar(Y)[2])-(mu[1]-mu[2]),S[p]*sqrt(frac(1,n[1])+frac(1,n[2])))) #mathematical annotation -see ?plotmath
legend("topright",c("Simulated","Theoretical"),col=c("Blue","Red"),lwd=4,lty=1:2,bty="n",title=title)# Legend #
invisible(list(w=w,summary=summary(w),sdw=sd(w),fun="T")) # some output to use if needed
}
myT2var1 = myTsim2(n1 = 10,n2=10, iter= 1000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 0.5)
MyT2var2 = myTsim2(n1 = 20,n2=10, iter= 1000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 0.5)
MyT2var3 = myTsim2(n1 = 50,n2=50, iter= 10000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 0.5)
MyT2var4 = myTsim2(n1 = 80,n2=50, iter= 10000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 0.5)
T2 = myTsim2(iter= 1000, ymax = 0.5)
hist(T2$w,col=rainbow(15), xlab = expression(paste(chi^2,"Value", sep=" "),las = 1))
Task 6
myFsim2<-function(n1=10,n2=14,sigma1=3,sigma2=2,mean1=5,mean2=10,iter=1000,ymax=0.9,...){
y1=rnorm(n1*iter,mean=mean1,sd=sigma1)# generate iter samples of size n1
y2=rnorm(n2*iter,mean=mean2,sd=sigma2)
data1.mat=matrix(y1,nrow=n1,ncol=iter,byrow=TRUE) # Each column is a sample size n1
data2.mat=matrix(y2,nrow=n2,ncol=iter,byrow=TRUE)
ssq1=apply(data1.mat,2,var) # ssq1 is s squared
ssq2=apply(data2.mat,2,var)
#spsq=((n1-1)*ssq1 + (n2-1)*ssq2)/(n1+n2-2) # pooled s squared
w=ssq1*sigma2^2/(ssq2*sigma1^2) #
hist(w,freq=FALSE, ylim=c(0,ymax), # Histogram with annotation
main=substitute(paste("Sample size = ",n[1]+n[2]," = ",n1+n2," statistic = ",F)),
xlab=paste("F Statistic",sep=""), las=1)
lines(density(w),col="Blue",lwd=3) # add a density plot
curve(df(x,n1-1,n2-1),xlim=c(0,6),add=TRUE,col="Red",lty=2,lwd=3) # add a theoretical curve
title=expression(F==frac(s[1]^2,s[2]^2)*frac(sigma[2]^2,sigma[1]^2)) #mathematical annotation -see ?plotmath
legend("topright",c("Simulated","Theoretical"),col=c("Blue","Red"),lwd=4,lty=1:2,bty="n",title=title)# Legend #
invisible(list(w=w,summary=summary(w),sd=sd(w),fun="F")) # some output to use if needed
}
myF1 = myFsim2(n1 = 10,n2=10, iter= 1000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 0.7)
myF2 = myFsim2(n1 = 20,n2=10, iter= 1000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 1)
myF3 = myFsim2(n1 = 50,n2=50, iter= 10000, mean1= 5,mean2 = 10 ,sigma1 = 4,sigma2 = 4,ymax = 2)
myF4 = myFsim2(n1 = 80,n2=50, iter= 10000, mean1= 3,mean2 = 5 ,sigma1 = 10,sigma2 = 10,ymax = 2)
FS2 = myFsim2(iter= 1000, ymax = 2)
hist(FS2$w,col=rainbow(15), xlab = expression(paste(chi^2,"Value", sep=" "),las = 1))
Task 7
library(MATH4753DPAI24)
##
## Attaching package: 'MATH4753DPAI24'
## The following object is masked _by_ '.GlobalEnv':
##
## mychisim
data("fire")
knitr::kable(head(fire))
| 3.4 |
26.2 |
| 1.8 |
17.8 |
| 4.6 |
31.3 |
| 2.3 |
23.1 |
| 3.1 |
27.5 |
| 5.5 |
36.0 |